Supersymmetry · PDF fileSupersymmetry Yuji Tachikawa An idea connecting Physics and...

Supersymmetry

Yuji Tachikawa

An idea connecting Physics and Mathematics

Kavli IPMU

• There are many Kavli Institutes around the world;

• The Kavli Institute I belong to has a special menu to offer: mathematics!

• Kavli IPMU =Kavli Institute for Physics and Mathematics of the Universe.

• So I decided to talk about something a bit mathematical.

Spectrum of researchers at Kavli IPMU

Pure mathematics

String theory

High energy particle physics

phenomenology

Cosmology

Astrophysics

High energy particle physicsexperiments

Spectrum of researchers at Kavli IPMU

Pure mathematics

String theory

High energy particle physics

phenomenology

Cosmology

Astrophysics

High energy particle physicsexperiments

I'm around here, so I'm not very representative ...

Supersymmetry

Symmetry

Supersymmetry

It’s an Extension of the concept of Symmetry

First, let me talk about a different Extension of

the concept of Symmetry

• There are many periodic tilings

From the Grammar of Ornament, (O. Jones, 1856)

6-fold symmetric

4-fold symmetric

• It's a simple mathematical fact that periodic tilings can only have 2, 3, 4, or 6-fold rotational symmetry.

tropic within experimental resolution. In addition, we studytwo pseudoternary compounds, (Y12xTbx)-Mg-Zn and(Y12xGdx)-Mg-Zn, and examine the effects of rare-earthconcentration and CEF ~crystalline electric field! splitting onT f . Results of electrical resistivity measurements are alsopresented. For comparison, the magnetic properties of aclosely related R-Mg-Zn crystalline phase are presented asan appendix.

II. SAMPLE PREPARATION

Single-grain samples of R-Mg-Zn quasicrystals (R5Y,Tb, Dy, Ho, Er! were grown from the ternary melt followingthe technique described in Ref. 3 In brief, this method in-volves the slow cooling of a ternary melt intersecting theprimary solidification surface of the quasicrystalline phase,as identified for Y-Mg-Zn by Langsdorf, Ritter and Assmus.6Large, single-grain samples can be grown by this technique,with volumes of up to 0.5 cm3. Following the growth, a briefdip in a dilute solution of nitric acid in methanol can be usedto remove any small amounts of undecanted flux or oxideslag that wet the surface. A photograph of a typical Ho-Mg-Zn quasicrystal is shown in Fig. 1, over a mm scale. Ascan be seen from Fig. 1, the quasicrystals grown by thistechnique have a dodecahedral growth habit, with clearlydefined pentagonal facets. It should be noted that theR-Mg-Zn quasicrystal samples grown by this technique areremarkably large and exceptionally well ordered.3Following the same growth procedure, samples with vari-

ous amounts of substitution of Tb and Gd for Y were alsoprepared, enabling investigation of magnetic and transportproperties as a function of the rare-earth concentration. Inparticular, the substitutions studied were (Y12xTbx)-Mg-Znwith x50.075, 0.15, 0.33, 0.50, 0.63, 0.75, and 0.88 and(Y12xGdx)-Mg-Zn with x50.075, 0.15, 0.50, and 0.60. TheGd substitutions are of additional interest because the pureGd-Mg-Zn quasicrystalline phase cannot be grown by thistechnique.3 In fact, x50.60 is the largest amount of Gd sub-stitution in Y-Mg-Zn that we have been able to successfully

incorporate as a single-phase quasicrystal. It remains a pos-sibility that the pure Gd-Mg-Zn phase is metastable, and canonly be prepared by rapid solidification techniques1 ~in con-trast to the slow cooling rate growth technique we employ!.Powder x-ray-diffraction patterns of crushed single grains

have narrow peaks, indicating a high degree of structuralorder, and all peaks can be indexed to the face-centeredicosahedral ~FCI! quasicrystalline phase ~second phases, ifpresent at all, are below a 2–5% level!.3 High-resolutiontransmission electron microscopy ~TEM! reveals very sharpdiffraction spots in selected area diffraction patterns~SADP’s!, with little or no evidence of phason strain seen inlattice images near Scherzer defocus.3 Electron microprobeanalysis of Tb-Mg-Zn quasicrystalline samples indicate acomposition of approximately Tb9Mg34Zn57,3 similar to thatestimated by other groups @R8Mg42Zn50 ~Refs. 1 and 7! andR9Mg30Zn61 ~Ref. 6!#.Clean, well-formed single grains were selected for mag-

netization measurements. Bars for electrical transport mea-surements were cut from single grains using a wire saw.Typical bar lengths were 2–3 mm with a width ~and thick-ness! of 0.5 mm. For icosahedral symmetry, the electricalresistivity tensor has only one independent component, andhence the orientation of the cut bars is not expected to effectthe results of the resistivity measurements.

III. EXPERIMENTAL METHODS

Both ac and dc magnetization were measured using com-mercial Quantum Design superconducting quantum interfer-ence device ~SQUID! magnetometers, in a variety of appliedmagnetic fields ~up to 55 000 Oe! and temperatures ~from 1.8to 350 K!. The dc magnetization was investigated for a rangeof applied fields, and for both zero-field-cooled ~zfc! andfield-cooled ~fc! histories. An applied field of 1000 Oe wasused to measure the temperature-dependent dc susceptibilityfor T.T f . The ac magnetization was measured between 1.8and 20 K in a steady applied field Hdc5100 Oe, with an acfield Hac52 Oe superimposed, and for a range of frequenciesfrom 1 to 1000 Hz. The angular dependence of the dc mag-netization was measured, using a rotating sample platform,with an angular resolution of 60.1°. All temperature-dependent magnetization measurements were made for in-creasing temperatures, and warming through the boilingpoint of liquid helium was treated carefully. After collectingdata from 1.8 to 4.4 K, the cryostat design required that thetemperature be cycled from 4.4 K to a higher temperature~usually 30 K! before zero-field-coding or field-cooling ~de-pending on the experiment! down to 4.6 K. Relaxation ef-fects for temperatures below the spin freezing transition ~seelater sections! can lead to an apparent discontinuity in the zfcmagnetization of Tb-Mg-Zn ~for which T f55.8 K! at 4.4 Kas a consequence of this technique.The low-frequency and low-field ac magnetic susceptibil-

ity was measured by a conventional mutual inductance tech-nique in the temperature range between 0.4 and 4 K. Mea-surements were made at frequencies 116 and 420 Hz andwith an excitation magnetic-field amplitude of 0.1 Oe. Alow-noise signal transformer was used to match the imped-ance of the pickup coil with that of the lock-in amplifier. Thein-phase component x8 and the out-of-phase component x9

FIG. 1. Photograph of a single-grain icosahedral Ho-Mg-Znquasicrystal grown from the ternary melt. Shown over a mm scale,the edges are 2.2 mm long. Note the clearly defined pentagonalfacets, and the dodecahedral morphology.

PRB 59 309MAGNETIC AND TRANSPORT PROPERTIES OF . . .

from Fischer et al., Phys. Rev. B 59 (1999) 308

• But there are crystalline materials that are 5-fold symmetric.

• Crystals are thought to be periodic arrays of atoms.

• But this cannot be completely periodic, since it is 5-fold symmetric.

• The way out: Quasi-periodicity.

• It’s infinitely close to being periodic, but not quite periodic.

• This quasi-periodic extension of the concept of symmetry was known to mathematicians, including Roger Penrose, since around 1970s.

• This was before real materials were found.

An example of Penrose tiling, from Wikipedia

• In fact it was known to Muslim artists in medieval times, as rediscovered in an article from 2007.

21. Materials and methods are available as supportingmaterial on Science Online.

22. A. Heger, N. Langer, Astron. Astrophys. 334, 210 (1998).23. A. P. Crotts, S. R. Heathcote, Nature 350, 683 (1991).24. J. Xu, A. Crotts, W. Kunkel, Astrophys. J. 451, 806 (1995).25. B. Sugerman, A. Crotts, W. Kunkel, S. Heathcote,

S. Lawrence, Astrophys. J. 627, 888 (2005).26. N. Soker, Astrophys. J., in press; preprint available online

(http://xxx.lanl.gov/abs/astro-ph/0610655)

27. N. Panagia et al., Astrophys. J. 459, L17 (1996).28. The authors would like to thank L. Nelson for providing

access to the Bishop/Sherbrooke Beowulf cluster (Elix3)which was used to perform the interacting windscalculations. The binary merger calculations wereperformed on the UK Astrophysical Fluids Facility.T.M. acknowledges support from the Research TrainingNetwork “Gamma-Ray Bursts: An Enigma and a Tool”during part of this work.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/315/5815/1103/DC1Materials and MethodsSOM TextTables S1 and S2ReferencesMovies S1 and S2

16 October 2006; accepted 15 January 200710.1126/science.1136351

Decagonal and Quasi-CrystallineTilings inMedieval Islamic ArchitecturePeter J. Lu1* and Paul J. Steinhardt2

The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns inmedieval Islamic architecture were conceived by their designers as a network of zigzagging lines,where the lines were drafted directly with a straightedge and a compass. We show that by 1200C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellationsof a special set of equilateral polygons (“girih tiles”) decorated with lines. These tiles enabledthe creation of increasingly complex periodic girih patterns, and by the 15th century, thetessellation approach was combined with self-similar transformations to construct nearly perfectquasi-crystalline Penrose patterns, five centuries before their discovery in the West.

Girih patterns constitute a wide-rangingdecorative idiom throughout Islamicart and architecture (1–6). Previous

studies of medieval Islamic documents de-scribing applications of mathematics in ar-chitecture suggest that these girih patternswere constructed by drafting directly a net-work of zigzagging lines (sometimes calledstrapwork) with the use of a compass andstraightedge (3, 7). The visual impact of thesegirih patterns is typically enhanced by rota-tional symmetry. However, periodic patternscreated by the repetition of a single “unit cell”motif can have only a limited set of rotationalsymmetries, which western mathematicians firstproved rigorously in the 19th century C.E.: Onlytwo-fold, three-fold, four-fold, and six-foldrotational symmetries are allowed. In particular,five-fold and 10-fold symmetries are expresslyforbidden (8). Thus, although pentagonal anddecagonal motifs appear frequently in Islamicarchitectural tilings, they typically adorn a unitcell repeated in a pattern with crystallographical-ly allowed symmetry (3–6).

Although simple periodic girih patterns in-corporating decagonal motifs can be constructedusing a “direct strapwork method” with astraightedge and a compass (as illustrated inFig. 1, A to D), far more complex decagonalpatterns also occur in medieval Islamic archi-tecture. These complex patterns can have unitcells containing hundreds of decagons and may

repeat the same decagonal motifs on severallength scales. Individually placing and draftinghundreds of such decagons with straightedgeand compass would have been both exceedinglycumbersome and likely to accumulate geometricdistortions, which are not observed.

On the basis of our examination of a largenumber of girih patterns decorating medievalIslamic buildings, architectural scrolls, and otherforms of medieval Islamic art, we suggest thatby 1200 C.E. there was an important break-through in Islamic mathematics and design: thediscovery of an entirely new way to conceptual-ize and construct girih line patterns as decoratedtessellations using a set of five tile types, whichwe call “girih tiles.” Each girih tile is decoratedwith lines and is sufficiently simple to be drawnusing only mathematical tools documented inmedieval Islamic sources. By laying the tilesedge-to-edge, the decorating lines connect toform a continuous network across the entiretiling. We further show how the girih-tileapproach opened the path to creating new typesof extraordinarily complex patterns, including anearly perfect quasi-crystalline Penrose patternon the Darb-i Imam shrine (Isfahan, Iran, 1453C.E.), whose underlying mathematics were notunderstood for another five centuries in theWest.

As an illustration of the two approaches,consider the pattern in Fig. 1E from the shrine ofKhwaja Abdullah Ansari at Gazargah in Herat,Afghanistan (1425 to 1429 C.E.) (3, 9), based ona periodic array of unit cells containing acommon decagonal motif in medieval Islamicarchitecture, the 10/3 star shown in Fig. 1A (seefig. S1 for additional examples) (1, 3–5, 10).Using techniques documented by medievalIslamic mathematicians (3, 7), each motif can

be drawn using the direct strapwork method(Fig. 1, A to D). However, an alternativegeometric construction can generate the samepattern (Fig. 1E, right). At the intersectionsbetween all pairs of line segments not within a10/3 star, bisecting the larger 108° angle yieldsline segments (dotted red in the figure) that, whenextended until they intersect, form three distinctpolygons: the decagon decorated with a 10/3 starline pattern, an elongated hexagon decoratedwith a bat-shaped line pattern, and a bowtiedecorated by two opposite-facing quadrilaterals.Applying the same procedure to a ∼15th-century pattern from the Great Mosque ofNayriz, Iran (fig. S2) (11) yields two additionalpolygons, a pentagon with a pentagonal starpattern, and a rhombus with a bowtie linepattern. These five polygons (Fig. 1F), whichwe term “girih tiles,” were used to construct awide range of patterns with decagonal motifs(fig. S3) (12). The outlines of the five girih tileswere also drawn in ink by medieval Islamicarchitects in scrolls drafted to transmit architec-tural practices, such as a 15th-century Timurid-Turkmen scroll now held by the Topkapi PalaceMuseum in Istanbul (Fig. 1G and fig. S4) (2, 13),providing direct historical documentation oftheir use.

The five girih tiles in Fig. 1F share severalgeometric features. Every edge of each polygonhas the same length, and two decorating linesintersect the midpoint of every edge at 72° and108° angles. This ensures that when the edges oftwo tiles are aligned in a tessellation, decoratinglines will continue across the common boundarywithout changing direction (14). Because bothline intersections and tiles only contain anglesthat are multiples of 36°, all line segments in thefinal girih strapwork pattern formed by girih-tiledecorating lines will be parallel to the sides ofthe regular pentagon; decagonal geometry isthus enforced in a girih pattern formed by thetessellation of any combination of girih tiles.The tile decorations have different internal ro-tational symmetries: the decagon, 10-fold sym-metry; the pentagon, five-fold; and the hexagon,bowtie, and rhombus, two-fold.

Tessellating these girih tiles provides severalpractical advantages over the direct strapworkmethod, allowing simpler, faster, and more ac-curate execution by artisans unfamiliar withtheir mathematical properties. A few full-sizegirih tiles could serve as templates to help po-sition decorating lines on a building surface,allowing rapid, exact pattern generation. More-

1Department of Physics, Harvard University, Cambridge,MA 02138, USA. 2Department of Physics and PrincetonCenter for Theoretical Physics, Princeton University,Princeton, NJ 08544, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

23 FEBRUARY 2007 VOL 315 SCIENCE www.sciencemag.org1106

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Science 315(2007)1106.

• It's a famous landmark in Morocco, visited by many, but nobody realized it's there until recently.

• e.g. from Al Attarine Madrasa, Fez, Morocco, 14c.

as pointed out in R. A. Al Ajlouni, Acta Crystalographica 2012 A68. Photo taken from http://toeuropeandbeyond.com/5-things-to-do-in-fes-morocco/

Quasi-periodicity: an Extension of the concept of Symmetry

Supersymmetry: an Extension of the concept of Symmetry

• It's again an extension of the concept of symmetry.

• It exchanges bosons and fermions.

This is an anniversary photo of Kavli IPMU from a few years ago.

If you exchange humans, you of course get different configurations!

If they were elementary particles, things are different.

ψ( )

=±ψ( )

The quantum state remains the same, up to an overall plus or minus sign.

ψ( )

=±ψ( )

They are bosons if the sign is plus.

ψ( )

=+ψ( )

They are fermions if the sign is minus.

ψ( )

=−ψ( )

• Bosons: photons, Higgs bosons ...

• Fermions: electrons, quarks, protons ...

• They are quite distinct !

• Ordinary symmetries can only map bosons to bosons and fermions to fermions.

• Supersymmetry maps fermions to bosons and bosons to fermions.

• Quite unusual, but not very super.

• Somehow the extravagant terminology used by the original authors stuck.

• Zero point energy of a bosonic harmonic oscillator= + ℏω/2

• Quantum bosonic fields (such as photons) ~ infinite number of bosonic harmonic oscillators~ infinite positive energy~ infinite gravitational attraction

• Not observed!

ω

• Zero point energy of a fermionic harmonic oscillator= − ℏω/2

• Quantum fermionic fields (such as electrons) ~ infinite number of fermionic harmonic oscillators~ infinite negative energy~ infinite gravitational repulsion

• Not observed!

ω

• Supersymmetry guarantees that bosons and fermions in our world are arranged so that infinite zero point energies cancel.

ω

• Supersymmetry guarantees that bosons and fermions in our world are arranged so that infinite zero point energies cancel.

ω

∞−∞=0 !

ω

SupersymmetricNot quite supersymmetric

ω

SupersymmetricNot quite supersymmetric

• in this case ∞−∞ is not exactly zero, but finite.

• Does the world work in this way?

• Nobody knows. One big motivation to build LHC was to see if this is the case.

from CERN webpage

• There's a theoretical suggestion that supersymmetry can be realized on the surface of a suitable material

• No experimental realization yet, but I think it is just a matter of time

References and Notes1. S. R. Kane, D. M. Gelino, Publ. Astron. Soc. Pac. 124,

323–328 (2012).2. J. F. Kasting, D. P. Whitmire, R. T. Reynolds, Icarus 101,

108–128 (1993).3. F. Selsis et al., Astron. Astrophys. 476, 1373–1387

(2007).4. R. K. Kopparapu et al., Astrophys. J. 765, 131 (2013).5. W. J. Borucki et al., Science 340, 587–590 (2013).6. J. C. Tarter et al., Astrobiology 7, 30–65 (2007).7. P. Kroupa, C. A. Tout, G. Gilmore, Mon. Not. R. Astron.

Soc. 262, 545–587 (1993).8. P. S. Muirhead et al., Astrophys. J. 750, L37 (2012).9. J. F. Rowe et al., Astrophys. J. 784, 45 (2014).

10. J. J. Lissauer et al., Astrophys. J. 784, 44 (2014).11. K. Mandel, E. Agol, Astrophys. J. 580, L171–L175 (2002).12. J. Goodman, J. Weare, Comm. App. Math. Comp. Sci. 5,

65–80 (2010).13. D. Foreman-Mackey, D. Hogg, D. Lang, J. Goodman,

Publ. Astron. Soc. Pac. 125, 306–312 (2013).14. M. Lopez-Morales, Astrophys. J. 660, 732–739 (2007).15. A. J. Bayless, J. A. Orosz, Astrophys. J. 651, 1155–1165

(2006).16. J. Irwin et al., Astrophys. J. 701, 1436–1449 (2009).17. M. Mayor, D. Queloz, Nature 378, 355–359 (1995).18. M. J. Holman et al., Science 330, 51–54 (2010).19. G. Torres, M. Konacki, D. Sasselov, S. Jha, Astrophys. J.

614, 979–989 (2004).20. T. Barclay et al., Astrophys. J. 768, 101 (2013).21. J. Wang, J. Xie, T. Barclay, D. Fischer, Astrophys. J. 783, 4

(2014).22. E.-M. David, E. V. Quintana, M. Fatuzzo, F. C. Adams,

Publ. Astron. Soc. Pac. 115, 825–836 (2003).23. E. D. Lopez, J. J. Fortney, N. Miller, Astrophys. J. 761, 59

(2012).24. J. J. Lissauer, Astrophys. J. 660, L149–L152 (2007).

25. S. N. Raymond, J. Scalo, V. S. Meadows, Astrophys. J.669, 606–614 (2007).

26. S. N. Raymond, R. Barnes, A. M. Mandell, Mon. Not. R.Astron. Soc. 384, 663–674 (2008).

27. E. Chiang, G. Laughlin, Mon. Not. R. Astron. Soc. 431,3444–3455 (2013).

28. C. Terquem, J. C. B. Papaloizou, Astrophys. J. 654,1110–1120 (2007).

29. C. Cossou, S. N. Raymond, A. Pierens, IAU Symp. 299,360–364 (2014).

30. S. M. Andrews, J. P. Williams, Astrophys. J. 671,1800–1812 (2007).

31. S. N. Raymond, C. Cossou, Mon. Not. R. Astron. Soc.(2014).

32. P. Hut, Astron. Astrophys. 99, 126–140 (1981).33. S. Ferraz-Mello, A. Rodriguez, H. Hussmann, Celestial

Mech. Dyn. Astron. 101, 171–201 (2008).34. R. Heller, J. Leconte, R. Barnes, Astron. Astrophys. 528,

A27 (2011).35. S. Udry et al., Astron. Astrophys. 469, L43–L47 (2007).36. P. von Paris et al., Astron. Astrophys. 522, A23 (2010).37. R. D. Wordsworth et al., Astrophys. J. 733, L48 (2011).

Acknowledgments: The authors working at NASA Amesthank the SETI Institute for hosting them during the U.S.government shutdown. E.V.Q. and J.F.R. acknowledge supportfrom the Research Opportunities in Space and Earth SciencesKepler Participating Scientist Program Grant NNX12AD21G.S.N.R.'s contribution was performed as part of the NASAAstrobiology Institute's Virtual Planetary Laboratory LeadTeam, supported by NASA under cooperative agreementno. NNA13AA93A. D.H. acknowledges support by anappointment to the NASA Postdoctoral Program at the AmesResearch Center, administered by Oak Ridge AssociatedUniversities through a contract with NASA, and the KeplerParticipating Scientist Program. The Center for Exoplanets and

Habitable Worlds is supported by the Pennsylvania StateUniversity, the Eberly College of Science, and the PennsylvaniaSpace Grant Consortium. F.S. acknowledges support from theEuropean Research Council (Starting Grant 209622: E3ARTHs).This paper includes data collected by the Kepler mission.Funding for the Kepler mission is provided by the NASAScience Mission directorate. This research also made use ofNASA's Astrophysics Data System. Some of the data presentedin this paper were obtained from the Mikulski Archive forSpace Telescopes (MAST). The Space Telescope Science Instituteis operated by the Association of Universities for Research inAstronomy, under NASA contract NAS5-26555. Support forMAST for non-Hubble Space Telescope data is provided by theNASA Office of Space Science via grant NNX13AC07G and byother grants and contracts. This research made use of theNASA Exoplanet Archive, which is operated by the CaliforniaInstitute of Technology, under contract with NASA under theExoplanet Exploration Program. The Gemini Observatory isoperated by the Association of Universities for Research inAstronomy, under a cooperative agreement with the NationalScience Foundation (NSF) on behalf of the Gemini partnership:NSF (United States), the National Research Council (Canada),the Comisión Nacional de Investigación Científica y Tecnológica(Chile), the Australian Research Council (Australia), theMinistério da Ciência, Tecnologia e Inovação (Brazil), and theMinisterio de Ciencia, Tecnología e Innovación Productiva(Argentina).

Supplementary Materialswww.sciencemag.org/content/344/6181/277/suppl/DC1Materials and MethodsFigs. S1 to S7Tables S1 and S2References (38–73)6 December 2013; accepted 12 March 201410.1126/science.1249403

Emergent Space-TimeSupersymmetry at the Boundaryof a Topological PhaseTarun Grover,1 D. N. Sheng,2 Ashvin Vishwanath3,4*

In contrast to ordinary symmetries, supersymmetry (SUSY) interchanges bosons and fermions.Originally proposed as a symmetry of our universe, it still awaits experimental verification.Here, we theoretically show that SUSY emerges naturally in condensed matter systems known astopological superconductors. We argue that the quantum phase transitions at the boundary oftopological superconductors in both two and three dimensions display SUSY when probed atlong distances and times. Experimental consequences include exact relations between quantitiesmeasured in disparate experiments and, in some cases, exact knowledge of the universalcritical exponents. The topological surface states themselves may be interpreted as arising fromspontaneously broken SUSY, indicating a deep relation between topological phases and SUSY.

Since the 1970s, space-time “supersymmetry”(SUSY) has been actively pursued by par-ticle physicists to attack the long-standing

hierarchy problem of fundamental forces (1–4).Unlike any other symmetry, SUSY interchangesbosons and fermions and, when applied twice,generates translations of space and time, which

ultimately leads to the conservation of momentumand energy (3). But despite sustained effort, SUSYhas yet to be experimentally established in nature.

Here, we theoretically show that certain con-densedmatter systems display phenomena of emer-gent SUSY; that is, space-time SUSY naturallyemerges as an accurate description of these sys-tems at low energy and at long distances, althoughthe microscopic ingredients are not supersym-metric. The physical systems we mainly considerare topological superconductors (TSCs) (5, 6), inwhich pairs of fermions, which may be electronsor fermionic atoms such as helium-3, pair togetherin a special way. The resulting state has an energy

gap to fermions in the bulk but gapless excitationsat the surface. We consider the quantum phase tran-sition at which the surface modes acquire a gap andestablish emergent supersymmetry D = 1 + 1 andD = 2 + 1 dimensional surfaces (Fig. 1) by using acombination of numerical and analytical tech-niques (D denotes the space-time dimensionality).

The study of SUSYat a phase transition pointwas initiated in (7), where it was shown that the1 + 1 dimensional tricritical Ising model, whichcan be accessed by tuning two parameters, is su-persymmetric. A few other proposals that realizeSUSY by fine-tuning two or more parametershave been made as well (8–11). Here, we requirethat SUSY be achievable by tuning only a singleparameter, akin to a conventional quantum criticalpoint (12). This is crucial for our results to be ex-perimentally realizable. We also require that ourtheory has full space-time SUSY rather than onlya limited “quantum-mechanical” SUSY (13). Thisautomatically ensures translation invariance in spaceand timeandwill lead to experimental consequences,as we discuss below. Perhaps most interesting, incontrast to the strategy adopted in (7), our approachis not restricted to 1 + 1 dimensional theories.

There has been an explosion of activity in thefield of topological phases since the discovery ofℤ2 topological insulators (TIs) (14–17). We willfocus on a set of closely related phases, the time-reversal invariant TSCs (5, 6), which include thewell-known B-phase of superfluid helium-3 (18).These phases exist in both two and three spatialdimensions (19, 20) and support Majorana modesat their boundary; the modes are protected by

1Kavli Institute for Theoretical Physics, University of California,Santa Barbara, CA 93106, USA. 2Department of Physics andAstronomy, California State University, North Ridge, CA 91330,USA. 3Department of Physics, University of California, Berkeley,CA 94720,USA. 4Materials Science Division, Lawrence BerkeleyNational Laboratory, Berkeley, California 94720, USA.

*Corresponding author. E-mail: [email protected]

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time-reversal symmetry from acquiring an ener-gy gap. Spontaneous breaking of this symmetryprovides a natural mechanism to gap them out.For example, electron-electron interactions at thesurface could lead tomagnetic order, which breakstime reversal symmetry. A natural question is howthe surface modes evolve as the magnetic ordersets in. We will see that space-time SUSY natu-rally emerges at the onset of magnetic order.

The D = 2 + 1 dimensional TSC, protectedby the time-reversal symmetry, provides the sim-plest setting to address this question. Whereas thebulk of the superconductor is gapped, the bound-ary, a Dedge = 1 + 1 dimensional system, containsa pair of Majorana modes cR,cL that propagate

in opposite directions. The aforementioned insta-bility of the edge may be described by introducingan Ising field f that changes sign under time re-versal. The action is given by

Sdþ1 ¼ ∫dt ddx 12c∂=cþ 1

2ð∂tfÞ2 þ

!

v2f2ð∇fÞ2 þ r

2f2 þ gfccþ uf4

"ð1Þ

with d ¼ 1, c ¼ ½cRcL&T , and we have used

the conventional Dirac gamma matrices for therelativistic fermion c ⋅ vf and u are, respectively,the velocity and self-interaction of f; g is thecoupling between the fermions and f, whereas

r is the tuning parameter for the transition. Thesymmetry-broken phase is characterized by f ≠ 0,which leads to a mass gap gf for the fermions.

The mode count in the action S is favorablefor N ¼ 1 SUSY in D = 1 + 1 (21), with thebosons f and Majorana fermions c as superpart-ners. We now show that this is indeed the case byusing a numerical simulation of a D = 1 + 1 lat-tice model that reproduces the action in Eq. 1 atlow energies. The model is given by

H ¼ −i∑j

1 − gmzjþ12

h icj cjþ1 þ Hb

where

Hb ¼∑j½J mzj−1=2m

zjþ1=2 − hmxjþ1=2& ð2Þ

Here, cj is a single Majorana fermion at site j,whereas the Ising spins mzjþ1=2 sit on bond centers.When the transverse magnetic field h ≫ 1, ⟨mz⟩ ¼ 0and lattice translation symmetry ensures that theMajorana fermions are gapless. As h decreases,at some point mz orders antiferromagnetically, lead-ing to a mass gap for the fermions through thecoupling g, reproducing the field theory in Eq. 1 atand near the critical point. J tunes the relativebare velocities between the boson and the fermionmodes, similar to vf in Eq. 1.

We numerically simulate a spin version ofEq. 2 by using the density matrix renormalizationgroup (DMRG)method (22).(Fig. 2). At larger h,a gapless phase is obtained that is separated by acritical line from a gapped ordered phase at smallh. To characterize the critical theory, considercrossing the phase boundary along fixed g, say,g ¼ 0:5, while monitoring the central charge c,which quantifies the amount of entanglement of a1 + 1 – D critical system (Fig. 2B). At small h, cis almost equal to zero, which indicates a gappedphase. At h > hc ≈ 1:62, the central charge satu-rates at c ≈ 0:5, indicating that the symmetry isrestored and a gapless Majorana mode is present.At the transition, h ¼ hc, we find c ≈ 0:7, whichprecisely corresponds to the SUSY tricritical Ising

Fig. 1. Supersymmetry in a 3D TSC. Ising magnetic fluctuations (denoted by red arrows) at theboundary couple to the Majorana fermions (blue cone). When the tuning parameter r < rc, the Ising spinsare ordered, leading to a gap for the Majorana fermions. The critical point that separates the two sides issupersymmetric, where bosons (Ising order parameter) and Majorana fermions transform into each other.

Fig. 2. Supersymmetryina2DTSC. (A) The phasediagram of the Hamilto-nian H in Eq. 2, whichrealizes theMajorana edgecoupled to Ising magnet-ic fluctuations in this sys-tem. At large h, the Isingspins disorder, and thecounterpropagating Ma-joranamodes (blue arrows)remain gapless. As h de-creases, the ordering ofIsing spins (red arrows)leads to a gap for the Ma-jorana modes. (B) The re-gion of the phase diagram indicated by the black arrows in (A). Plotted isthe central charge c as a function of h for fixed g = 0:5,J = 1. Forh > hc (= 1:62), c equals 1/2, consistent with gapless Majorana modes; forh < hc, c is 0, which indicates the gapped phase. At the critical point, c is 7/10,

which corresponds to supersymmetric tricritical Ising point. (Inset) von Neumannentropy S and the Renyi entropies S2,S3 at the critical point, which were usedto deduce the central charge. L is the system size, and l is the size of theentanglement subsystem.

www.sciencemag.org SCIENCE VOL 344 18 APRIL 2014 281

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• So, it's not clear supersymmetry is in nature or not.

• I am often asked, "do you think LHC find supersymmetry?" or "what would you do if LHC would not find supersymmetry?"

• Well, I do not care.

• I'm no Christian nor Buddhist, but I like reading books on Christian Theology and Buddhist Buddhology.

• Just in the same way, I like thinking about supersymmetry. It's interesting to me.

• For me, supersymmetry is interesting mostly because of its connection with mathematics.

1983: Donaldson (a mathematician) found that by studying the Yang-Mills equation describing quantum chromodynamics very carefully, you can understand the mathematical properties of four-dimensional manifolds in exquisite detail.

1983

Donaldson theory

Mathematics Theoretical Physics

1988: Witten (a physicist) found that Donaldson's results can be thought of as a statement about supersymmetric Yang-Mills.

1983

SupersymmetricYang-Mills

1988

Donaldson theory


(Witten)

1994: Seiberg and Witten found physically that supersymmetric Yang-Mills reduces to supersymmetric Maxwell.

SupersymmetricMaxwell

1994

1983


1988

Donaldson theory


(Witten)

(Seiberg-Witten)

1994~: Therefore, to study four-dimensional manifolds, you don't have to study the Yang-Mills equation which is rather difficult; You just have to study the Maxwell equation. This brought a sudden revolutionary development in this area of mathematics.

1983

Donaldson theory


Monopole equation

DrasticSimplification

1994~


1994


1988 (Witten)

(Seiberg-Witten)


1994


1988

2002 Nekrasov (a physicist) reformulated this derivation in a way understandable to mathematicians

Theoretical Physics

(Witten)

(Seiberg-Witten)

2003

That reformulation was then proved by mathematicians Nakajima, Yoshioka;Braverman, Etingof; Nekrasov, Okounkov

2002 Nekrasov (a physicist) reformulated this derivation in a way understandable to mathematicians


1994


1988 (Witten)

(Seiberg-Witten)

2003

That reformulation was then proved by mathematicians Nakajima, Yoshioka;Braverman, Etingof; Nekrasov, Okounkov

2009 Based on these results, Alday, Gaiotto and I thought more about physics and found a mathematical conjecture


1994


1988 (Witten)

(Seiberg-Witten)

2009 Based on these results, Alday, Gaiotto and I thought more about physics and found a mathematical conjecture

2012 The conjecture was proven by mathematicians, Shiffman and Vasserot; Maulik and Okounkov


1994


1988 (Witten)

(Seiberg-Witten)

Theoretical Physics

Access the reality by means of experiments

Come up with random ideas.

Mathematics

Access the platonic reality by means of proofs

Come up with random ideas.

Supersymmetric "Physics"

Experimental Physics

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